MODULI SPACES OF CONNECTIONS ON A RIEMANN SURFACE

Size: px
Start display at page:

Download "MODULI SPACES OF CONNECTIONS ON A RIEMANN SURFACE"

Transcription

1 MODULI SPACES OF CONNECTIONS ON A RIEMANN SURFACE INDRANIL BISWAS AND VICENTE MUÑOZ Abstract. Let E be a holomorphic vector bundle over a compact connected Riemann surface X. The vector bundle E admits a holomorphic projective connection if and only if for every holomorphic direct summand F of E of positive rank, the equality degree(e)/rank(e) = degree(f )/rank(f ) holds. Fix a point x 0 in X. There is a logarithmic connection on E, singular over x 0 with residue d n Id E x0 if and only if the equality degree(e)/rank(e) = degree(f )/rank(f ) holds. Fix an integer n 2, and also fix an integer d coprime to n. Let M(n, d) denote the moduli space of logarithmic SL(n, C) connections on X singular of x 0 with residue d n Id. The isomorphism class of the variety M(n, d) determines the isomorphism class of the Riemann surface X. 1. Introduction Let X be a compact connected Riemann surface. Let E be a holomorphic vector bundle over X of rank n. A holomorphic connection on E is given by locally defined holomorphic trivializations of E such that all the transition functions are locally constant maps to GL(n, C). We recall that if {U i } i I is a covering of X by open subsets and ϕ i : E Ui U i C n are holomorphic isomorphisms, then for an ordered pair (i, j) I I, the transition function g i,j : U i U j GL(n, C) is the unique function satisfying the identity g i,j ϕ i = ϕ j. Therefore, if E is equipped with a holomorphic connection, then it makes sense to talk of locally constant holomorphic sections of E. A holomorphic projective connection on E is given by locally defined holomorphic trivializations of E such that all the transition functions g i,j project to locally constant maps under the projection GL(n, C) PGL(n, C). Therefore, if E is equipped with a holomorphic projective connection, then the projective bundle P(E) has locally defined holomorphic trivializations such that all the transition functions are locally constant maps to PGL(n, C). Fix a point x 0 X. Set X := X \ {x 0 } to be the complement. A logarithmic connection on E singular at x 0 is a holomorphic connection on E X which, locally around x 0, has a holomorphic connection matrix with a single pole at x 0. Given a logarithmic 2000 Mathematics Subject Classification. 14H60, 14C30. 1

2 2 I. BISWAS AND V. MUÑOZ connection on E singular at x 0, the behavior of the flat sections near x 0 are captured by what is called the residue of the connection. The residue is a linear endomorphism of the fiber E x0. If the residue of a logarithmic connection on E singular at x 0 is a scalar multiple of the identity automorphism of E x0, then the logarithmic connection gives a holomorphic projective connection on E. Here we shall consider logarithmic connections whose residue is a scalar multiple of the identity automorphism. We investigate some properties of holomorphic projective connections and logarithmic connections. Especially we address the question of existence of such connections on a given holomorphic vector bundle. In the last section we explain a recent result of the authors on the moduli spaces of logarithmic connections. 2. Connections and projective connections Let X be a compact connected Riemann surface; alternatively, X is an irreducible smooth projective curve defined over the field of complex numbers. The complexified tangent bundle T R X R C decomposes as T R X R C = T 1,0 X T 0,1 X into (1, 0) and (0, 1) components. The dual line bundle (T 1,0 X) will also be denoted by Ω 1,0 X or K X, and the complex line bundle (T 0,1 X) will also be denoted by Ω 0,1 X. A holomorphic vector bundle over X is a C vector bundle of rank n whose transition functions are given by holomorphic maps to GL(n, C). The usual Dolbeault operator -operator on locally defined smooth functions over X give a Dolbeault operator E on a holomorphic vector bundle E. This operator E is a first order differential operator that sends smooth sections of E to those of E Ω 0,1 X. We note that the line bundles T 1,0 X and K X have natural holomorphic structures. The trivial holomorphic line bundle X C will also be denoted by O X. Definition 2.1. Let E be a holomorphic vector bundle over X of rank n. A holomorphic connection on E is a first order holomorphic differential operator D : E E K X satisfying the Leibniz identity, which says that D(fs) = fd(s) + df s, where f (respectively, s) is a locally defined holomorphic function (respectively, holomorphic section of E). Associated to a holomorphic connection D there is a C connection : C (E) C (E T X),

3 MODULI SPACE OF CONNECTIONS 3 where C (E) denotes the sheaf of smooth sections of the vector bundle E. Using the isomorphism Γ(E) = O(E) O C, where O(E) is the sheaf of holomorphic sections of E, we have (fs) = fd(s) + df s, for s a holomorphic section of E and f C (X). In other terms, we define = D + E on holomorphic sections, and then we extend to C sections. Let D be a holomorphic connection on E. The curvature D D of D is a holomorphic section of the vector bundle End(E) Ω 2,0 X. Since dim C X = 1, we have Ω 2,0 X = 0. Hence a holomorphic connection on a Riemann surface is automatically flat. If E has a holomorphic connection, then degree(e) = 0 [At]. Actually, the connection is also flat, since its curvature is F = D E + E D, which is zero on holomorphic sections, hence F = 0. Considering a local flat trivialization, the constant sections (in the local trivialization) are automatically holomorphic, hence there are local flat holomorphic trivializations of E. Note also that any flat connection on a C vector bundle E gives rise to a holomorphic structure on E together with a holomorphic connection. For this, we only need to take flat trivializations and declare them to be holomorphic. Let E be a holomorphic vector bundle over X and F a direct summand of E. This means that F is a holomorphic subbundle of E, and there exists a holomorphic subbundle F of E such that the natural homomorphism F F E is an isomorphism. Let ι : F E be the inclusion map. Fix a complement F of F as above. Using the natural isomorphism F F E we get a projection q : E F. If D is a holomorphic connection on E, then it is easy to see that the composition F ι E D E K X q Id F K X is a holomorphic connection on F. Hence if E admits a holomorphic connection, then each direct summand of E also admits a holomorphic connection. So, if E admits a holomorphic connection, then the degree of each direct summand of E is zero. A theorem due to Atiyah and Weil says that the converse is also true. Theorem 2.2 ([At], [We]). A holomorphic vector bundle E over X admits a holomorphic connection if and only if the degree of each direct summand of E is zero. In particular, any indecomposable bundle E admits a holomorphic connection. A holomorphic vector bundle E of rank n over X admits a holomorphic connection if and only if we can choose (holomorphic) local trivializations of E over a covering of

4 4 I. BISWAS AND V. MUÑOZ X (by open sets) such that all the transition functions are locally constant functions to GL(n, C). This follows from the flatness of. Recall that the Atiyah bundle At(E) associated to a holomorphic vector bundle E is the holomorphic vector bundle associated to the sheaf of first order holomorphic differential operators on E. The Atiyah exact sequence for E is (2.1) 0 End(E) At(E) T 1,0 X 0 where the last arrow is the symbol map. Note that End(E) is identified with the zeroth order differential operators. Therefore, a holomorphic connection on E corresponds to a holomorphic splitting of the Atiyah exact sequence in (2.1). Now let P E be the holomorphic principal GL(n, C) bundle over X defined by E. So P E is the space of all bases in the fibers of E. Let O X End(E) be the line subbundle given by the sheaf of endomorphisms of E of the type s f s. Therefore, from (2.1) we have the exact sequence of vector bundles (2.2) 0 End(E)/O X At(E)/O X T 1,0 X 0 over X. This exact sequence is the Atiyah bundle for the principal PGL(n, C) bundle over X corresponding to E. We note that this principal PGL(n, C) is the quotient space P E /C, where C is considered as a subgroup of GL(n, C) through scalar multiplications of C n. Also, note that the quotient vector bundle End(E)/O X is identified with the subbundle of ad(e) End(E) defined by trace zero endomorphisms. Indeed, the natural projection of the subbundle ad(e) to the quotient bundle End(E)/O X is an isomorphism. Definition 2.3. A holomorphic projective connection on E is a holomorphic splitting of the exact sequence in (2.2). By a projective connection we will always mean a holomorphic projective connection. It follows that a holomorphic vector bundle E admits a projective connection if we can choose holomorphic local trivializations of E over a covering of X (by open sets) such that images of all the transition functions are locally constant under the projection GL(n, C) PGL(n, C). This means that they are of the form f Id n n times a locally constant function, with f being some nonzero holomorphic function. To see this, consider the splitting of the exact sequence (2.2) given by the projective connection, and take a covering by open sets U α. Over each U α, take an arbitrary splitting of At(E) At(E)/O X. This produces a holomorphic connection D α on U α, and hence a flat trivialization of E Uα. Since D α D β = f αβ Id n n on U α Uβ, we have that the transition functions of E (with respect to the chosen trivializations) are of the form f Id n n times a locally constant function. Assume that the vector bundle E admits a projective connection. Let E PGL be the principal PGL(n, C) bundle given by E. We noted earlier that E PGL = P E /C, where P E is the principal GL(n, C) bundle given by E. Giving a projective connection on E is

5 MODULI SPACE OF CONNECTIONS 5 equivalent to giving a holomorphic connection on the principal PGL(n, C) bundle E PGL, that is, a family of flat trivializations of E PGL such that the transition functions are locally constant functions with values in PGL(n, C). Note that this also can be expressed in terms of giving trivializations of the projective bundle P(E) whose transition functions are locally constant functions in PGL(n, C). Using the commutative diagram of vector bundles 0 End(E) At(E) T 1,0 X 0 0 End(E)/O X At(E)/O X T 1,0 X 0 over X, a holomorphic connection on E induces a projective connection on E. However, the converse is not true in the sense that a vector bundle admitting a projective connection need not admit a holomorphic connection. If E admits a projective connection and degree(e) = 0 then the projective connection is induced by a holomorphic connection. To see this, take a covering by open sets U α with a holomorphic connection D α on each U α, as before. Since D α D β = f αβ Id n n on U α Uβ, we have that f αβ Ω 1 X (U α Uβ ) form a cocycle, and hence define a class in H 1 (X, K X ) = C. Therefore there exists g α Ω 1 X (U α) such that if we modify D α to D α = D α +g α Id n n, then the corresponding D α D β is a constant multiple of the identity. This gives transition functions for E which are locally constant functions in GL(n, C). The following theorem classifies all holomorphic vector bundles over X that admit a projective connection. Theorem 2.4. Let E be a holomorphic vector bundle over X of rank n and degree d. Then the following two statements are equivalent: (1) The vector bundle E admits a projective connection. (2) For any direct summand F E, the equality holds. degree(f ) rank(f ) Proof. Assume that the vector bundle E admits a projective connection. Let E PGL be the principal PGL(n, C) bundle given by E. The projective connection on E gives a holomorphic connection on E PGL. The adjoint vector bundle over X for the principal PGL(n, C) bundle E PGL is ad(e). Therefore ad(e) admits a family of trivializations with transitions functions locally constant functions on PGL(n, C). This means that ad(e) admits a projective connection. Since the degree of ad(e) is zero, the vector bundle ad(e) admits a holomorphic connection. = d n Let D be a holomorphic connection on ad(e). Assume that E = F 1 F 2.

6 6 I. BISWAS AND V. MUÑOZ We will show that D induces a holomorphic connection on the vector bundle Hom(F 1, F 2 ). For this, first note that Hom(F 1, F 2 ) is a direct summand of ad(e). Indeed, we have a holomorphic decomposition (2.3) ad(e) = Hom(F 1, F 2 ) Hom(F 2, F 1 ) (End(F 1 ) End(F 2 ))/O X of ad(e) into a direct sum of vector bundles; here O X is considered as a subbundle of End(F 1 ) End(F 2 ) by sending any locally defined holomorphic function f to f(id F1 + Id F2 ). Hom(F 1, F 2 ) is a direct summand of the bundle ad(e), which admits a holomorphic connection. Hence Theorem 2.2 implies that (2.4) degree(hom(f 1, F 2 )) = n 1 d 2 d 1 n 2 = 0, where n i (respectively, d i ) is the rank (respectively, degree) of F i, i = 1, 2. Since from (2.4), it follows immediately that degree(e) = d 1 + d 2 = d, degree(f i ) rank(f i ) = d n. Therefore, the first statement in the theorem implies the second statement. To prove the converse, we begin with a lemma. Lemma 2.5. Let E 1 and E 2 are two holomorphic vector bundles over X, both admitting projective connections. If degree(e 1 ) = degree(e 2) rank(e 1 ) rank(e 2 ), then E 1 E2 also admits a projective connection. Proof. Since both E 1 and E 2 admit projective connections, the vector bundle E 1 E2 also admits a projective connection. On the other hand, the condition that degree(e 1 ) rank(e 1 ) = degree(e 2) rank(e 2 ) implies that degree(e 1 E2 ) = 0. This condition together with the condition that E 1 E2 admits a projective connection imply that E 1 E2 admits a holomorphic connection. Similarly, E 2 E1 admits a holomorphic connection. Now using the natural decomposition End(E 1 E 2 ) = End(E 1 ) End(E 2 ) (E 1 E 2 ) (E 2 E 1 ) it follows that E 1 E2 also admits a projective connection, and the proof of the lemma is complete. Continuing with the proof of the theorem, assume that for each direct summand F of E. degree(f ) rank(f ) = d n

7 MODULI SPACE OF CONNECTIONS 7 We will prove that E admits a projective connection. This will be done by imitating the Atiyah s proof in [At] of the criterion for the existence of a holomorphic connection. As before, let E PGL denote the holomorphic principal PGL(n, C) bundle over X defined by E. Let (2.5) 0 ad(e) At(E PGL ) T 1,0 X 0 be the Atiyah exact sequence for the principal PGL(n, C) bundle E PGL (see [At, page 187, Theorem 1]), where At(E PGL ) is the Atiyah bundle for E PGL. The exact sequence of vector bundles in (2.5) coincides with the exact sequence in (2.2). A projective connection on E is equivalent to a holomorphic connection on the principal PGL(n, C) bundle E PGL. Let (2.6) a(e) H 1 (X, K X ad(e)) be the obstruction class for holomorphic splitting of (2.5). We note that the vector bundle ad(e) is self-dual, that is, ad(e) = ad(e). Indeed, the trace map ad(e) ad(e) O X that sends s t to trace(s t) identifies ad(e) with ad(e). Therefore, the Serre duality says that Let H 1 (X, K X ad(e)) = H 0 (X, ad(e)). (2.7) b(e) H 0 (X, ad(e)) be the linear form corresponding to the cohomology class a(e), constructed in (2.6), by the Serre duality. In view of Lemma 2.5, it suffices to prove the theorem under the assumption that the vector bundle E is indecomposable. Assume that E is indecomposable. Then any section s H 0 (X, ad(e)) is actually nilpotent [At, page 201, Proposition 16]. In other words, the endomorphism s gives a holomorphic filtration of subbundles 0 = E 0 E 1 E 2 E k = E such that s(e i ) = E i 1 for all 1 i k. More precisely, E i is the subbundle of E generated by the kernel of the i-fold composition s i = s s. Now, for a nilpotent endomorphism s of E we know that b(e)(s) = 0, where b(e) is constructed in (2.7) [At, page 202, Proposition 18(ii)]. Since the functional b(e) vanishes on H 0 (X, ad(e)), we have b(e) = 0. Therefore, the obstruction class a(e)

8 8 I. BISWAS AND V. MUÑOZ in (2.6) vanishes, and hence E PGL admits a holomorphic connection. This completes the proof of the theorem. 3. Logarithmic connections As before, X will denote a compact connected Riemann surface of genus g, where g 2. Fix a point x 0 X. The holomorphic line bundle over X defined by the divisor x 0 will be denoted by O X (x 0 ). Definition 3.1. Let E be a holomorphic vector bundle over X. A logarithmic connection on E singular over x 0 is a holomorphic differential operator satisfying the Leibniz identity D : E E O X (x 0 ) K X (3.1) D(fs) = fd(s) + df s, where f (respectively, s) is a locally defined holomorphic function (respectively, holomorphic section of E). Note that a logarithmic connection on E singular over x 0 produces a holomorphic connection on E over X \ {x 0 }. Let U be a chart around x 0 with local holomorphic coordinate z. Then (3.2) D(s) = ds + A(z)s dz z, for any local holomorphic section s of End(E) and a holomorphic connection matrix A(z). The curvature of a holomorphic connection on E is a holomorphic two form with values in End(E) O X (x 0 ). Since a Riemann surface does not have nonzero holomorphic two forms, any logarithmic connection on a Riemann surface is flat. A differential operator that satisfies the Leibniz identity (3.1) is clearly of order one. The above condition that a logarithmic connection D satisfies the Leibniz identity is evidently equivalent to the condition that the symbol of the first order differential operator D coincides with the section Id E H 0 (X, O X (x 0 ) End(E)), where Id E denotes the identity automorphism of E. The Poincaré adjunction formula says the following: Let D be a smooth hypersurface on a smooth variety Z, and let L denote the line bundle over Z defined by the divisor D. Then the restriction of the line bundle L to D is canonically identified with the normal bundle of D. See [GH] for a proof. Therefore, the Poincaré adjunction formula says that the fiber over x 0 of the line bundle O X (x 0 ) is identified with the holomorphic tangent space T x0 X. Using this isomorphism between O X (x 0 ) x0 and T x0 X, the fiber (K X OX (x 0 )) x0 is identified with C.

9 MODULI SPACE OF CONNECTIONS 9 We will now recall the definition of the important notion of residue of a logarithmic connection. Let D be a logarithmic connection on a vector bundle E over X, which is singular over x 0. Let v E x0 be any vector in the fiber of E over the point x 0. Let v be a holomorphic section of E defined around x 0 such that v(x 0 ) = v. Consider D( v)(x 0 ) (K X O X (x 0 )) x0 C E x0 = C C E x0 = E x0. Note that if v = 0, then D( v) is a (locally defined) section of the subsheaf E K X E O X (x 0 ) K X. So, in that case the above evaluation D( v)(x 0 ) E x0 vanishes. Consequently, we have a well defined endomorphism Res(D, x 0 ) End(E x0 ) that sends any v E x0 to D( v)(x 0 ) E x0. This endomorphism Res(D, x 0 ) is called the residue of the logarithmic connection D at the point x 0. See [De1] for more details. If D is a logarithmic connection on E singular over x 0 and θ H 0 (X, End(E) K X ), then the differential operator D + θ is also a logarithmic connection on E singular over x 0. Furthermore, we have Res(D, x 0 ) = Res(D + θ, x 0 ). Conversely, if D and D are two logarithmic connections on E regular on X \ {x 0 } with (3.3) Res(D, x 0 ) = Res(D, x 0 ), then D = D + θ, where θ H 0 (X, End(E) K X ). In this way the space of all logarithmic connections D on the given vector bundle E, regular on X\{x 0 } and satisfying (3.3), with D fixed, is an affine space for the vector space H 0 (X, K X End(E)). Let E be a holomorphic vector bundle over X of rank n, and let D be a logarithmic connection on E, regular on X \ {x 0 }, satisfying the residue condition (3.4) Res(D, x 0 ) = d n Id E x0, where d C. Consider the nonsingular flat connection on the complement X \ {x 0 } defined by D. The above condition on the residue implies that the monodromy around x 0 of this flat connection is the n n diagonal matrix with exp(2π 1d/n) as the diagonal entries [De1, page 79, Proposition 3.11]. From the expression of the degree of E in terms of the residue of D it follows immediately that degree(e) = d; the last sentence of Corollary B.3 in [EV, page 186] gives the expression of the first Chern class in terms of the trace of the residue. In particular, d in (3.4) must be an integer. We will now relate logarithmic connections with projective connections discussed in Section 2. Let E be a holomorphic vector bundle over X of rank n and degree d. Let D be a logarithmic connection on E, regular on X \ {x 0 }, satisfying the residue condition in

10 10 I. BISWAS AND V. MUÑOZ (3.4). The logarithmic connection D gives a holomorphic connection on the vector bundle E X\{x0 } over the open subset X \ {x 0 } X. This connection on E X\{x0 } induces a projective connection on the PGL(n, C) principal bundle E PGL X\{x0 } over X \{x 0 } X, understood as a family of (holomorphic) trivializations with transition functions which are locally constant in PGL(n, C). The residue of D is in the center of M(n, C) (the Lie algebra of GL(n, C)); its projection to sl(n, C) vanishes. Therefore, the projective connection of E PGL X\{x0 } extends across x 0 as a regular projective connection. Hence we have proved the following lemma: Lemma 3.2. Let E be a holomorphic vector bundle over X such that E admits a logarithmic connection D regular on X \ {x 0 } such that Res(D, x 0 ) = d n Id E x0. Then the vector bundle E admits a projective connection. The following theorem is the analog of Theorem 2.4 for logarithmic connections. Theorem 3.3. Let E be a holomorphic vector bundle over X of rank n and degree d. Then the following two statements are equivalent: (1) The vector bundle E admits a logarithmic connection regular on X \ {x 0 } whose residue at x 0 is d n Id E x0. (2) For any direct summand F E, the equality holds. degree(f ) rank(f ) Proof. Let D be a logarithmic connection on the vector bundle E, regular on X\{x 0 }, with residue Res(D, x 0 ) = d n Id E x0. From Lemma 3.2 we know that E admits a projective connection. Therefore, from Theorem 2.4 it follows that for every direct summand F of E. To prove the converse, assume that (3.5) degree(f ) rank(f ) degree(f ) rank(f ) for every direct summand F of E. Therefore, E admits a projective connection (see Theorem 2.4). Fix a projective connection D P on E. Let = d n = d n = d n (3.6) γ : X X

11 MODULI SPACE OF CONNECTIONS 11 be a ramified Galois covering of degree n with the property that that the map γ is totally ramified over the point x 0 X. (The map γ is allowed to have ramifications over points in X \{x 0 }.) The condition that γ is totally ramified over x 0 means that there is a unique point y 0 X such that γ(y 0 ) = x 0. Consider the vector bundle (3.7) V := O X( dy 0 ) O X γ E over X, where γ is the map in (3.6), and y 0 is the unique point with γ(y 0 ) = x 0. The projective bundle P(V ) over X is canonically identified with the projective bundle γ P(E). Therefore, the projective connection D P on E gives a projective connection γ D P on V. Note that degree(v ) = 0. Therefore, the condition that V admits a projective connection implies that V admits a holomorphic connection. Fix a holomorphic connection D on V. Let Γ denote the Galois group for the Galois covering γ in (3.6). Note that the Galois action of Γ on X has a canonical lift to the vector bundle V. Consider D = 1 h D #Γ which is a holomorphic connection on V (as the space of holomorphic connections on V is an affine space, the average of D over the action of the elements of Γ is again a holomorphic connection on V ). It is now straight forward to check that the Galois invariant connection D on V descends to a logarithmic connection on E. This descended logarithmic connection is regular outside x 0, and its residue at x 0 is d Id n E x0. This completes the proof of the theorem. Theorem 2.4 and Theorem 3.3 together have the following corollary: Corollary 3.4. Let E be a holomorphic vector bundle over X of rank n and degree d. Then the following three statements are equivalent: h Γ (1) The vector bundle E admits a logarithmic connection regular on X \ {x 0 } whose residue at x 0 is d n Id E x0. (2) The vector bundle E admits a projective connection. (3) For any direct summand F E, the equality degree(f ) rank(f ) = d n holds.

12 12 I. BISWAS AND V. MUÑOZ 4. Moduli space of logarithmic connections Henceforth, we will assume the rank n and the degree d are mutually coprime. Let M D (n) denote the moduli space of all logarithmic connections (E, D) over X, where E is any holomorphic vector bundle of rank n and degree d and D is a logarithmic connection on E, regular on X \ {x 0 }, satisfying the residue condition Res(D, x 0 ) = d n Id E x0. (See [Si1], [Si2], [Ni] for the construction of this moduli space.) So M D (n) parametrizes isomorphism classes of pairs of the form (E, D), where E is any rank n holomorphic vector bundle of degree d and D is a logarithmic connection on E regular on X \ {x 0 } with Res(D, x 0 ) = d Id n E x0. We note that every logarithmic connection over X occurring in the moduli space M D (n) is irreducible. Indeed, for (E, D) M D (n), if F E is a holomorphic subbundle invariant by the connection D, i.e. D(F ) F O(x 0 ) K X, then D induces a logarithmic connection D F on F which is regular on X \ {x 0 } and Res(D F, x 0 ) = d n Id F x0. Therefore, degree(f ) = d rank(f ) n. This contradicts the assumption that d and n are mutually coprime if F is a proper subbundle of E. Since every logarithmic connection over X occurring in the moduli space M D (n) is irreducible, the variety M D (n) is smooth; singular points of a moduli space of connections correspond to reducible connections. The variety M D (n) is known to be irreducible. The dimension of M D (n) is 2(n 2 (g 1) + 1). Consider the holomorphic line bundle L := O X (d x 0 ) over X. The de Rham differential f df, defines a logarithmic connection D L on L which is regular on X \ {x 0 }, and (4.1) Res(D L, x 0 ) = d Id Lx0. Let (4.2) M D (L) M D (n) be the subvariety parametrizing isomorphism classes of logarithmic connections (E, D) M D (n) such that n E = L, and the logarithmic connection on L induced by D using an isomorphism n E L coincides with the logarithmic connection D L in (4.1). Note that a logarithmic connection D on E induces a logarithmic connection on n E. Since any two holomorphic isomorphisms between n E and L differ by a constant scalar,

13 MODULI SPACE OF CONNECTIONS 13 the connection of L given by the induced connection on n E using an isomorphism n E L is independent of the choice of the isomorphism. The subset M D (L) in (4.2) is an irreducible smooth closed subvariety of dimension 2(n 2 1)(g 1). We will show that the biholomorphism class of both M D (L) and M D (n) are independent of the complex structure of X. Let X := X \ {x 0 } be the complement. Fix a point x X. The point x 0 gives a conjugacy class in the fundamental group π 1 (X, x ) as follows. Let f : D X be an orientation preserving embedding of the closed unit disk D := {z C z 2 1} into the Riemann surface X such that f(0) = x 0. The free homotopy class of the map D = S 1 X obtained by restricting f to the boundary of D is independent of the choice of f. The orientation of D coincides with the anti clockwise rotation around x 0. Any free homotopy class of oriented loops in X gives a conjugacy class in π 1 (X, x ). Let γ denote the orbit in π 1 (X, x ), for the conjugation action of π 1 (X, x ) on itself, defined by the above free homotopy class of oriented loops associated to x 0. Let (4.3) Hom 0 (π 1 (X, x ), GL(n, C)) Hom(π 1 (X, x ), GL(n, C)) be the space of all homomorphisms from the fundamental group π 1 (X, x ) to GL(n, C) satisfying the condition that the image of γ (the free homotopy class defined above) is exp(2π 1d/n) I n n. It may be noted that since exp(2π 1d/n) I n n is in the center of SL(n, C), a homomorphism sends the orbit γ in π 1 (X, x ) (for the adjoint action of π 1 (X, x ) on itself) to exp(2π 1d/n) I n n if and only if there is an element in the orbit which is mapped to exp(2π 1d/n) I n n. Take any homomorphism ρ Hom 0 (π 1 (X, x ), GL(n, C)). Let (V, ) be the flat vector bundle or rank n over X given by ρ. Therefore V is a holomorphic vector bundle on X. The monodromy of along the oriented loop γ is exp(2π 1d/n) I n n. Using the logarithm 2π 1d/n I n n of the monodromy, the vector bundle V over X extends to a holomorphic vector bundle V over X, and furthermore, the connection on V extends to a logarithmic connection on the vector bundle V over X such that (V, ) M D (n), where M D (n) is the moduli space of logarithmic connections defined earlier (see [Ma, p. 159, Theorem 4.4]). Since GL(n, C) is an algebraic group defined over the field of complex numbers, and π 1 (X, x ) is a finitely presented group, the representation space Hom(π 1 (X, x ), SL(n, C)) is a complex algebraic variety in a natural way. The conjugation action of GL(n, C) on itself induces an action of GL(n, C) on Hom 0 (π 1 (X, x ), GL(n, C)). The action of any T GL(n, C) on Hom 0 (π 1 (X, x ), GL(n, C)) sends any homomorphism ρ to the

14 14 I. BISWAS AND V. MUÑOZ homomorphism π 1 (X, x ) GL(n, C) defined by β T 1 ρ(β)t. Let (4.4) R g := Hom 0 (π 1 (X, x ), GL(n, C))/GL(n, C) be the quotient space for this action. The algebraic structure of Hom 0 (π 1 (X, x ), GL(n, C)) induces an algebraic structure on the quotient R g. The scheme R g is an irreducible smooth quasiprojective variety of dimension 2(n 2 1)(g 1) + 2 defined over C. The isomorphism class of this variety R g is independent of the complex structure of the topological surface X. The isomorphism class depends only on the integers g, n and d. On the other hand, the moduli space M D (n) is canonically biholomorphic to R g. Therefore, the biholomorphism class of the moduli space M D (n) is independent of the complex structure of X; the biholomorphism class depends only on the integers g, n and d. Replacing GL(n, C) by the algebraic subgroup SL(n, C) above, we have an algebraic irreducible smooth quasiprojective variety S g := Hom 0 (π 1 (X, x ), SL(n, C))/SL(n, C), which is biholomorphic to M D (L). We conclude that the biholomorphism class of the moduli space M D (L) defined in (4.2) is also independent of the complex structure of X. 5. The second intermediate Jacobian of the moduli space In this section we recall some results of [BM]. A holomorphic vector bundle E over X is called stable if for every nonzero proper subbundle F E, the inequality holds. degree(f ) rank(f ) < degree(e) rank(e) Let N X denote the moduli space parametrizing all stable vector bundles E over X with rank(e) = n and n E = L = O X (d x 0 ). The moduli space N X is an irreducible smooth projective variety of dimension (n 2 1)(g 1) defined over C. Let M D (L) be the moduli space defined in (4.2). Let (5.1) U M D (L) be the Zariski open subset parametrizing all (E, D) such that the underlying vector bundle E is stable. The openness of this subset follows from [Ma]. Let (5.2) Φ : U N X denote the forgetful map that sends any (E, D) to E. By Theorem 3.3, any E N X admits a logarithmic connection D such that (E, D) M D (L), since any E N X is indecomposable. Therefore, the projection Φ in (5.2) is

15 MODULI SPACE OF CONNECTIONS 15 surjective. Furthermore, Φ makes U an affine bundle over N X. More precisely, U is a torsor over N X for the holomorphic cotangent bundle T N X. This means that the fibers of the vector bundle T N X act freely transitively on the fibers of Φ [BR, p. 786]. Since M D (L) is irreducible, and U is nonempty, the subset U M D (L) is Zariski dense. The following lemma is proved in [BM]: Lemma 5.1. Let Z := M D (L) \ U be the complement of the Zariski open dense subset. The codimension of the Zariski closed subset Z in M X is at least (n 1)(g 2) + 1. For any i 0, the i th cohomology of a complex variety with coefficients in Z is equipped with a mixed Hodge structure [De2], [De3]. The following proposition is proved in [BM]. Proposition 5.2. The mixed Hodge structure H 3 (M D (L), Z) is pure of weight three. More precisely, the mixed Hodge structure H 3 (M D (L), Z) is isomorphic to the Hodge structure H 3 (N X, Z), where N X is the moduli space of stable vector bundles introduced at the beginning of this section. Let (5.3) J 2 (M D (L)) := H 3 (M D (L), C)/(F 2 H 3 (M D (L), C) + H 3 (M D (L), Z)) be the intermediate Jacobian of the mixed Hodge structure H 3 (M D (L)) (see [Ca, p. 110]). The intermediate Jacobian of any mixed Hodge structure is a generalized torus [Ca, p. 111]. Let J 2 (N X ) := H 3 (N X, C)/(F 2 H 3 (N X, C) + H 3 (N X, Z)) be the intermediate Jacobian for H 3 (N X, Z), which is a complex torus. The following proposition is proved in [BM]. Proposition 5.3. The intermediate Jacobian J 2 (M D (L)) is isomorphic to J 2 (N X ), which is isomorphic to the Jacobian Pic 0 (X) of the Riemann surface X. The homology H 1 (J 2 (M D (L)), Z) has a natural skew symmetric pairing which we will describe below. We first note that from Proposition 5.3, and the fact that H 3 (N X, Z) is torsionfree, it follows that (5.4) H 1 (J 2 (M D (L)), Z) = H 3 (M D (L), Z) = H 3 (N X, Z). Also, we have Fix a generator For any H 2 (M D (L), Z) = H 2 (N X, Z) = Z. α H 2 (M D (L), Z). (5.5) θ H 2(n 2 1)(g 1)(M D (L), Z),

16 16 I. BISWAS AND V. MUÑOZ we have a skew symmetric pairing (5.6) B θ : 2 H 3 (M D (L), Z) Z defined by B θ (ω 1, ω 2 ) = (ω 1 ω 2 α (n2 1)(g 1) 3 ) θ Z. The following theorem is proved in [BM]. Theorem 5.4. There exists a homology class θ as in (5.5) such that the pairing B θ in (5.6) is nonzero. Take any θ such that B θ is nonzero. Using the isomorphism in (5.4), B θ defines a nonzero pairing B θ : 2 H1 (J 2 (M D (L)), Z) Z. The pair (J 2 (M D (L)), B θ ) is isomorphic to Pic 0 (X) equipped with a multiple of the canonical principal polarization on Pic 0 (X) given by the class of a theta line bundle. Given a multiple of a principal polarization on an abelian variety, there is a unique way to recover the principal polarization from it. Therefore, from Theorem 5.4 it follows immediately that the isomorphism class of (Pic 0 (X), Θ), where Θ is the canonical polarization on Pic 0 (X), is determined by the isomorphism class of the variety M D (L). The Torelli theorem says that the isomorphism class of the principally polarized abelian variety (Pic 0 (X), Θ) determines the curve X up to an isomorphism. Thus Theorem 5.4 give the following corollary: Corollary 5.5. The isomorphism class of the variety M D (L) uniquely determines the isomorphism class of the curve X. This is in contrast with the fact that the biholomorphism class of the complex manifold M D (L) is independent of the complex structure of the Riemann surface X (see the end of Section 4). References [At] M. F. Atiyah, Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85 (1957), [BM] I. Biswas and V. Muñoz, Torelli theorem for the moduli spaces of connections on a Riemann surface. Preprint. [BR] I. Biswas and N. Raghavendra, Line bundles over a moduli space of logarithmic connections on a Riemann surface. Geom. Funct. Anal. 15 (2005), [Ca] J. Carlson, Extensions of mixed Hodge structures, in: Journées de géométrie algébrique d Angers Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn Germantown, Md., 1980, pp [De1] P. Deligne, Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Math. 163, Springer Verlag, Berlin, [De2] P. Deligne, Théorie de Hodge, II. Inst. Hautes Études Sci. Publ. Math. 40 (1971), [De3] P. Deligne, Théorie de Hodge, III. Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5 77.

17 MODULI SPACE OF CONNECTIONS 17 [EV] H. Esnault and E. Viehweg, Logarithmic de Rham complex and vanishing theorems. Invent. Math. 86 (1986), [GH] P. A. Griffiths and J. Harris, Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York, [Ma] M. Maruyama, Openness of a family of torsion free sheaves. Jour. Math. Kyoto Univ. 16 (1976), [Ni] N. Nitsure, Moduli of semistable logarithmic connections. Jour. Amer. Math. Soc. 6 (1993), [Si1] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. 79 (1994), [Si2] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. 80 (1994), [We] A. Weil, Généralisation des fonctions abéliennes. Jour. Math. Pures Appl. 17 (1938), School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay , India address: indranil@math.tifr.res.in Departamento de Matemáticas, Consejo Superior de Investigaciones Científicas, Serrano 113 bis, Madrid, Spain address: vicente.munoz@imaff.cfmac.csic.es

A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface

A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface Documenta Math. 111 A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface Indranil Biswas Received: August 8, 2013 Communicated by Edward Frenkel

More information

NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD

NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD INDRANIL BISWAS Abstract. Our aim is to review some recent results on holomorphic principal bundles over a compact Kähler manifold.

More information

HOMOMORPHISMS OF VECTOR BUNDLES ON CURVES AND PARABOLIC VECTOR BUNDLES ON A SYMMETRIC PRODUCT

HOMOMORPHISMS OF VECTOR BUNDLES ON CURVES AND PARABOLIC VECTOR BUNDLES ON A SYMMETRIC PRODUCT PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 9, September 2012, Pages 3017 3024 S 0002-9939(2012)11227-4 Article electronically published on January 24, 2012 HOMOMORPHISMS OF VECTOR

More information

arxiv:math/ v1 [math.ag] 18 Oct 2003

arxiv:math/ v1 [math.ag] 18 Oct 2003 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI

More information

APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD. 1. Introduction

APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD. 1. Introduction APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Given a flat Higgs vector bundle (E,, ϕ) over a compact

More information

Stability of the Tangent Bundle of the Wonderful Compactification of an Adjoint Group

Stability of the Tangent Bundle of the Wonderful Compactification of an Adjoint Group Documenta Math. 1465 Stability of the Tangent Bundle of the Wonderful Compactification of an Adjoint Group Indranil Biswas and S. Senthamarai Kannan Received: April 3, 2013 Communicated by Thomas Peternell

More information

TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2)

TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2) TitleOn manifolds with trivial logarithm Author(s) Winkelmann, Jorg Citation Osaka Journal of Mathematics. 41(2) Issue 2004-06 Date Text Version publisher URL http://hdl.handle.net/11094/7844 DOI Rights

More information

The Atiyah bundle and connections on a principal bundle

The Atiyah bundle and connections on a principal bundle Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata

More information

Holomorphic line bundles

Holomorphic line bundles Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

More information

A CRITERION FOR A DEGREE-ONE HOLOMORPHIC MAP TO BE A BIHOLOMORPHISM. 1. Introduction

A CRITERION FOR A DEGREE-ONE HOLOMORPHIC MAP TO BE A BIHOLOMORPHISM. 1. Introduction A CRITERION FOR A DEGREE-ONE HOLOMORPHIC MAP TO BE A BIHOLOMORPHISM GAUTAM BHARALI, INDRANIL BISWAS, AND GEORG SCHUMACHER Abstract. Let X and Y be compact connected complex manifolds of the same dimension

More information

September 27, :51 WSPC/INSTRUCTION FILE biswas-loftin. Hermitian Einstein connections on principal bundles over flat affine manifolds

September 27, :51 WSPC/INSTRUCTION FILE biswas-loftin. Hermitian Einstein connections on principal bundles over flat affine manifolds International Journal of Mathematics c World Scientific Publishing Company Hermitian Einstein connections on principal bundles over flat affine manifolds Indranil Biswas School of Mathematics Tata Institute

More information

ON THE MAPPING CLASS GROUP ACTION ON THE COHOMOLOGY OF THE REPRESENTATION SPACE OF A SURFACE

ON THE MAPPING CLASS GROUP ACTION ON THE COHOMOLOGY OF THE REPRESENTATION SPACE OF A SURFACE PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 6, June 1996 ON THE MAPPING CLASS GROUP ACTION ON THE COHOMOLOGY OF THE REPRESENTATION SPACE OF A SURFACE INDRANIL BISWAS (Communicated

More information

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

More information

Contributors. Preface

Contributors. Preface Contents Contributors Preface v xv 1 Kähler Manifolds by E. Cattani 1 1.1 Complex Manifolds........................... 2 1.1.1 Definition and Examples.................... 2 1.1.2 Holomorphic Vector Bundles..................

More information

Cohomology jump loci of quasi-projective varieties

Cohomology jump loci of quasi-projective varieties Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)

More information

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally

More information

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle on a general hypersurface

More information

1. Algebraic vector bundles. Affine Varieties

1. Algebraic vector bundles. Affine Varieties 0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

More information

Period Domains. Carlson. June 24, 2010

Period Domains. Carlson. June 24, 2010 Period Domains Carlson June 4, 00 Carlson - Period Domains Period domains are parameter spaces for marked Hodge structures. We call Γ\D the period space, which is a parameter space of isomorphism classes

More information

Math 797W Homework 4

Math 797W Homework 4 Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

More information

Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

More information

THE VORTEX EQUATION ON AFFINE MANIFOLDS. 1. Introduction

THE VORTEX EQUATION ON AFFINE MANIFOLDS. 1. Introduction THE VORTEX EQUATION ON AFFINE MANIFOLDS INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show

More information

HODGE THEORY, SINGULARITIES AND D-MODULES

HODGE THEORY, SINGULARITIES AND D-MODULES Claude Sabbah HODGE THEORY, SINGULARITIES AND D-MODULES LECTURE NOTES (CIRM, LUMINY, MARCH 2007) C. Sabbah UMR 7640 du CNRS, Centre de Mathématiques Laurent Schwartz, École polytechnique, F 91128 Palaiseau

More information

Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016

Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016 Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces

More information

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

More information

CANONICAL BUNDLE FORMULA AND VANISHING THEOREM

CANONICAL BUNDLE FORMULA AND VANISHING THEOREM CANONICAL BUNDLE FORMULA AND VANISHING THEOREM OSAMU FUJINO Abstract. In this paper, we treat two different topics. We give sample computations of our canonical bundle formula. They help us understand

More information

A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES

A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES INDRANIL BISWAS, AJNEET DHILLON, JACQUES HURTUBISE, AND RICHARD A. WENTWORTH Abstract. Let be a compact connected Riemann surface. Fix a positive integer

More information

1 Moduli spaces of polarized Hodge structures.

1 Moduli spaces of polarized Hodge structures. 1 Moduli spaces of polarized Hodge structures. First of all, we briefly summarize the classical theory of the moduli spaces of polarized Hodge structures. 1.1 The moduli space M h = Γ\D h. Let n be an

More information

arxiv:alg-geom/ v1 29 Jul 1993

arxiv:alg-geom/ v1 29 Jul 1993 Hyperkähler embeddings and holomorphic symplectic geometry. Mikhail Verbitsky, verbit@math.harvard.edu arxiv:alg-geom/9307009v1 29 Jul 1993 0. ntroduction. n this paper we are studying complex analytic

More information

3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X). 3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

More information

Pacific Journal of Mathematics

Pacific Journal of Mathematics Pacific Journal of Mathematics THE MODULI OF FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES Eugene Z. Xia Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 THE MODULI OF

More information

RIEMANN S INEQUALITY AND RIEMANN-ROCH

RIEMANN S INEQUALITY AND RIEMANN-ROCH RIEMANN S INEQUALITY AND RIEMANN-ROCH DONU ARAPURA Fix a compact connected Riemann surface X of genus g. Riemann s inequality gives a sufficient condition to construct meromorphic functions with prescribed

More information

Coherent sheaves on elliptic curves.

Coherent sheaves on elliptic curves. Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of

More information

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Benson Farb and Mark Kisin May 8, 2009 Abstract Using Margulis s results on lattices in semisimple Lie groups, we prove the Grothendieck-

More information

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection

More information

HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

More information

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are C-linear. 1.

More information

MODULI SPACES OF CURVES

MODULI SPACES OF CURVES MODULI SPACES OF CURVES SCOTT NOLLET Abstract. My goal is to introduce vocabulary and present examples that will help graduate students to better follow lectures at TAGS 2018. Assuming some background

More information

ON A THEOREM OF CAMPANA AND PĂUN

ON A THEOREM OF CAMPANA AND PĂUN ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified

More information

Quaternionic Complexes

Quaternionic Complexes Quaternionic Complexes Andreas Čap University of Vienna Berlin, March 2007 Andreas Čap (University of Vienna) Quaternionic Complexes Berlin, March 2007 1 / 19 based on the joint article math.dg/0508534

More information

mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending

mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending 2. The canonical divisor In this section we will introduce one of the most important invariants in the birational classification of varieties. Definition 2.1. Let X be a normal quasi-projective variety

More information

Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

More information

arxiv: v2 [math.ag] 5 Jun 2018

arxiv: v2 [math.ag] 5 Jun 2018 A Brief Survey of Higgs Bundles 1 21/03/2018 Ronald A. Zúñiga-Rojas 2 Centro de Investigaciones Matemáticas y Metamatemáticas CIMM Universidad de Costa Rica UCR San José 11501, Costa Rica e-mail: ronald.zunigarojas@ucr.ac.cr

More information

Higgs Bundles and Character Varieties

Higgs Bundles and Character Varieties Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character

More information

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY NICK MCCLEEREY 0. Complex Differential Forms Consider a complex manifold X n (of complex dimension n) 1, and consider its complexified tangent bundle T C

More information

DELIGNE S THEOREM ON THE SEMISIMPLICITY OF A POLARIZED VHS OVER A QUASIPROJECTIVE BASE

DELIGNE S THEOREM ON THE SEMISIMPLICITY OF A POLARIZED VHS OVER A QUASIPROJECTIVE BASE DELIGNE S THEOREM ON THE SEMISIMPLICITY OF A POLARIZED VHS OVER A QUASIPROJECTIVE BASE ALEX WRIGHT 1. Introduction The purpose of this expository note is to give Deligne s proof of the semisimplicity of

More information

HARTSHORNE EXERCISES

HARTSHORNE EXERCISES HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

More information

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

More information

Polarized K3 surfaces of genus 18 and 20

Polarized K3 surfaces of genus 18 and 20 Polarized K3 surfaces of genus 18 and 20 Dedicated to Professor Hisasi Morikawa on his 60th Birthday Shigeru MUKAI A surface, i.e., 2-dimensional compact complex manifold, S is of type K3 if its canonical

More information

Homological mirror symmetry via families of Lagrangians

Homological mirror symmetry via families of Lagrangians Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants

More information

Notes on the geometry of Lagrangian torus fibrations

Notes on the geometry of Lagrangian torus fibrations Notes on the geometry of Lagrangian torus fibrations U. Bruzzo International School for Advanced Studies, Trieste bruzzo@sissa.it 1 Introduction These notes have developed from the text of a talk 1 where

More information

Hodge Theory of Maps

Hodge Theory of Maps Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

More information

A Note on Dormant Opers of Rank p 1 in Characteristic p

A Note on Dormant Opers of Rank p 1 in Characteristic p A Note on Dormant Opers of Rank p 1 in Characteristic p Yuichiro Hoshi May 2017 Abstract. In the present paper, we prove that the set of equivalence classes of dormant opers of rank p 1 over a projective

More information

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be

More information

Oral exam practice problems: Algebraic Geometry

Oral exam practice problems: Algebraic Geometry Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n

More information

Stable bundles on CP 3 and special holonomies

Stable bundles on CP 3 and special holonomies Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M

More information

The Hitchin map, local to global

The Hitchin map, local to global The Hitchin map, local to global Andrei Negut Let X be a smooth projective curve of genus g > 1, a semisimple group and Bun = Bun (X) the moduli stack of principal bundles on X. In this talk, we will present

More information

THE JACOBIAN OF A RIEMANN SURFACE

THE JACOBIAN OF A RIEMANN SURFACE THE JACOBIAN OF A RIEMANN SURFACE DONU ARAPURA Fix a compact connected Riemann surface X of genus g. The set of divisors Div(X) forms an abelian group. A divisor is called principal if it equals div(f)

More information

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current Author, F., and S. Author. (2015) Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current, International Mathematics Research Notices, Vol. 2015, Article ID rnn999, 7 pages. doi:10.1093/imrn/rnn999

More information

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N 74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the

More information

Canonical bundle formula and vanishing theorem

Canonical bundle formula and vanishing theorem RIMS Kôkyûroku Bessatsu Bx (200x), 000 000 Canonical bundle formula and vanishing theorem By Osamu Fujino Abstract In this paper, we treat two different topics. We give sample computations of our canonical

More information

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.

More information

Porteous s Formula for Maps between Coherent Sheaves

Porteous s Formula for Maps between Coherent Sheaves Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

More information

EXISTENCE THEORY FOR HARMONIC METRICS

EXISTENCE THEORY FOR HARMONIC METRICS EXISTENCE THEORY FOR HARMONIC METRICS These are the notes of a talk given by the author in Asheville at the workshop Higgs bundles and Harmonic maps in January 2015. It aims to sketch the proof of the

More information

Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman

Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman Commun. Math. Phys. 211, 359 363 2000) Communications in Mathematical Physics Springer-Verlag 2000 Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman Hélène snault 1, I-Hsun

More information

VARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN

VARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN VARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN Abstract. For any field k and integer g 3, we exhibit a curve X over k of genus g such that X has no non-trivial automorphisms over k. 1. Statement

More information

Dirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017

Dirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017 Dirac Operator Göttingen Mathematical Institute Paul Baum Penn State 6 February, 2017 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. The Riemann-Roch theorem 5. K-theory

More information

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON THREE DIMENSIONAL HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON THREE DIMENSIONAL HYPERSURFACES ARITHMETICALLY COHEN-MACAULAY BUNDLES ON THREE DIMENSIONAL HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle

More information

THE HITCHIN FIBRATION

THE HITCHIN FIBRATION THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus

More information

BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt, CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

More information

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree

More information

Some Remarks on Prill s Problem

Some Remarks on Prill s Problem AFFINE ALGEBRAIC GEOMETRY pp. 287 292 Some Remarks on Prill s Problem Abstract. N. Mohan Kumar If f : X Y is a non-constant map of smooth curves over C and if there is a degree two map π : X C where C

More information

NOTES ON DIVISORS AND RIEMANN-ROCH

NOTES ON DIVISORS AND RIEMANN-ROCH NOTES ON DIVISORS AND RIEMANN-ROCH NILAY KUMAR Recall that due to the maximum principle, there are no nonconstant holomorphic functions on a compact complex manifold. The next best objects to study, as

More information

Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

More information

PRINCIPAL BUNDLES OVER PROJECTIVE MANIFOLDS WITH PARABOLIC STRUCTURE OVER A DIVISOR

PRINCIPAL BUNDLES OVER PROJECTIVE MANIFOLDS WITH PARABOLIC STRUCTURE OVER A DIVISOR Tohoku Math. J. 53(2001), 337-367 PRINCIPAL BUNDLES OVER PROJECTIVE MANIFOLDS WITH PARABOLIC STRUCTURE OVER A DIVISOR Dedicated to the memory of Krishnamurty Guruprasad VlKRAMAN BALAJI, INDRANIL BISWAS

More information

Useful theorems in complex geometry

Useful theorems in complex geometry Useful theorems in complex geometry Diego Matessi April 30, 2003 Abstract This is a list of main theorems in complex geometry that I will use throughout the course on Calabi-Yau manifolds and Mirror Symmetry.

More information

Introduction To K3 Surfaces (Part 2)

Introduction To K3 Surfaces (Part 2) Introduction To K3 Surfaces (Part 2) James Smith Calf 26th May 2005 Abstract In this second introductory talk, we shall take a look at moduli spaces for certain families of K3 surfaces. We introduce the

More information

NON FORMAL COMPACT MANIFOLDS WITH SMALL BETTI NUMBERS

NON FORMAL COMPACT MANIFOLDS WITH SMALL BETTI NUMBERS NON FORMAL COMPACT MANIFOLDS WITH SMALL BETTI NUMBERS MARISA FERNÁNDEZ Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain E-mail:

More information

F. LAYTIMI AND D.S. NAGARAJ

F. LAYTIMI AND D.S. NAGARAJ REMARKS ON RAMANUJAM-KAWAMATA-VIEHWEG VANISHING THEOREM arxiv:1702.04476v1 [math.ag] 15 Feb 2017 F. LAYTIMI AND D.S. NAGARAJ Abstract. In this article weproveageneralresult on anef vector bundle E on a

More information

Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H.

Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H. Monodromy of the Dwork family, following Shepherd-Barron 1. The Dwork family. Consider the equation (f λ ) f λ (X 0, X 1,..., X n ) = λ(x n+1 0 + + X n+1 n ) (n + 1)X 0... X n = 0, where λ is a free parameter.

More information

COMPLEX ALGEBRAIC SURFACES CLASS 4

COMPLEX ALGEBRAIC SURFACES CLASS 4 COMPLEX ALGEBRAIC SURFACES CLASS 4 RAVI VAKIL CONTENTS 1. Serre duality and Riemann-Roch; back to curves 2 2. Applications of Riemann-Roch 2 2.1. Classification of genus 2 curves 3 2.2. A numerical criterion

More information

Vanishing theorems and holomorphic forms

Vanishing theorems and holomorphic forms Vanishing theorems and holomorphic forms Mihnea Popa Northwestern AMS Meeting, Lansing March 14, 2015 Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. Holomorphic one-forms and

More information

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

More information

EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION

EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION JAN CHRISTIAN ROHDE Introduction By string theoretical considerations one is interested in Calabi-Yau manifolds since Calabi-Yau 3-manifolds

More information

LECTURE 11: SOERGEL BIMODULES

LECTURE 11: SOERGEL BIMODULES LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing

More information

PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES

PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What

More information

PARABOLIC HIGGS BUNDLES AND Γ-HIGGS BUNDLES

PARABOLIC HIGGS BUNDLES AND Γ-HIGGS BUNDLES J. Aust. Math. Soc. 95 (2013), 315 328 doi:10.1017/s1446788713000335 PARABOLIC HIGGS BUNDLES AND Γ-HIGGS BUNDLES INDRANIL BISWAS, SOURADEEP MAJUMDER and MICHAEL LENNOX WONG (Received 31 August 2012; accepted

More information

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

More information

Riemann Surfaces and Algebraic Curves

Riemann Surfaces and Algebraic Curves Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1

More information

GENERIC ABELIAN VARIETIES WITH REAL MULTIPLICATION ARE NOT JACOBIANS

GENERIC ABELIAN VARIETIES WITH REAL MULTIPLICATION ARE NOT JACOBIANS GENERIC ABELIAN VARIETIES WITH REAL MULTIPLICATION ARE NOT JACOBIANS JOHAN DE JONG AND SHOU-WU ZHANG Contents Section 1. Introduction 1 Section 2. Mapping class groups and hyperelliptic locus 3 Subsection

More information

MODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION

MODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION Masuda, K. Osaka J. Math. 38 (200), 50 506 MODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION KAYO MASUDA (Received June 2, 999). Introduction and result Let be a reductive complex algebraic

More information

LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS

LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere

More information

THE QUANTUM CONNECTION

THE QUANTUM CONNECTION THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

More information

Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

More information

Mathematical Research Letters 3, (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION. Hanspeter Kraft and Frank Kutzschebauch

Mathematical Research Letters 3, (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION. Hanspeter Kraft and Frank Kutzschebauch Mathematical Research Letters 3, 619 627 (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION Hanspeter Kraft and Frank Kutzschebauch Abstract. We show that every algebraic action of a linearly reductive

More information

Symplectic varieties and Poisson deformations

Symplectic varieties and Poisson deformations Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the

More information

Algebraic geometry versus Kähler geometry

Algebraic geometry versus Kähler geometry Algebraic geometry versus Kähler geometry Claire Voisin CNRS, Institut de mathématiques de Jussieu Contents 0 Introduction 1 1 Hodge theory 2 1.1 The Hodge decomposition............................. 2

More information

THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

More information

Cohomology jump loci of local systems

Cohomology jump loci of local systems Cohomology jump loci of local systems Botong Wang Joint work with Nero Budur University of Notre Dame June 28 2013 Introduction Given a topological space X, we can associate some homotopy invariants to

More information

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree

More information